Question: Simplify; express your answer in exponential form. Assume $a\neq 0, z\neq 0$. $\dfrac{{(az^{2})^{5}}}{{(a^{-4}z^{-5})^{5}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(az^{2})^{5} = (a)^{5}(z^{2})^{5}}$ On the left, we have ${a}$ to the exponent ${5}$ . Now ${1 \times 5 = 5}$ , so ${(a)^{5} = a^{5}}$ Apply the ideas above to simplify the equation. $\dfrac{{(az^{2})^{5}}}{{(a^{-4}z^{-5})^{5}}} = \dfrac{{a^{5}z^{10}}}{{a^{-20}z^{-25}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{5}z^{10}}}{{a^{-20}z^{-25}}} = \dfrac{{a^{5}}}{{a^{-20}}} \cdot \dfrac{{z^{10}}}{{z^{-25}}} = a^{{5} - {(-20)}} \cdot z^{{10} - {(-25)}} = a^{25}z^{35}$